Take a complex harmonic signal with a component with amplitude A, angular frequency ω and phase p
x(t) = A0 * exp(j * (ω0 * t + p0))
and use it as an input to a system with transfer function H(j ω). The output will match the following equation:
y(t) = A1 * exp(j * (ω0 * t + p1)).
Note that the fundamental frequency has not changed, only the amplitude and the phase of the response changed as it went through the system. The transfer function describes for frequency ω0,
A1/A0 = | H(j * ω0) |
and
p1 - p0 = angle( H(j * ω0) ).